Abstract
Understanding and operating on multi-digit numbers is a critical step in the development of mathematical skills and concepts. Empirical and computational modeling evidence suggest that multi-digit number magnitude is processed in a componential manner, decomposed into units, tens, hundreds, etc. However, computational models evaluating whether such componential processing generalizes to multi-digit \textit{arithmetic} have so far relied on discrete symbolic inputs rather than on noisy magnitude representations. Accordingly, we developed a hierarchical neural network model of multi-digit addition based on componential processing of noisy single-digit magnitude representations. To evaluate model performance we implemented two training curricula: (i) an all-at-once curriculum and (ii) a step-by-step curriculum, where single-digit addition is trained before progressing to multi-digit problems (following school curricula). Under the training conditions tested, only the step-by-step curriculum allowed the model to learn multi-digit addition. Once trained this way, the model reproduced the carry-over and problem-size effects observed in reaction times of children and adults, both across problem categories and across individual items. These findings generalize the idea of decomposed processing of multi-digit number magnitude to arithmetic and highlight the importance of structured, incremental learning in both cognitive modeling and education.